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Consistency and matching without replacement. (2019). Savje, Fredrik .
In: Papers.
RePEc:arx:papers:1907.07288.

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  1. Abadie, A. & Imbens, G. W. (2006). Large sample properties of matching estimators for average treatment effects. Econometrica, 74(1), 235–267.

  2. Abadie, A. & Imbens, G. W. (2011). Bias-corrected matching estimators for average treatment effects. Journal of Business & Economic Statistics, 29(1), 1–11.

  3. Abadie, A. & Imbens, G. W. (2012). A martingale representation for matching estimators. Journal of the American Statistical Association, 107(498), 833–843.

  4. Abadie, A. & Imbens, G. W. (2016). Matching on the estimated propensity score. Econometrica, 84(2), 781–807.

  5. Beyer, K. S., Goldstein, J., Ramakrishnan, R., & Shaft, U. (1999). When is “nearest neighbor” meaningful? In Proceedings of the 7th International Conference on Database Theory, ICDT ’99 (pp. 217–235). Berlin: Springer.
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  6. Cochran, W. G. & Rubin, D. B. (1973). Controlling bias in observational studies: A review. Sankhyā: The Indian Journal of Statistics, Series A, 35(4), 417–446.
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  7. Heckman, J. J., Ichimura, H., & Todd, P. (1998). Matching as an econometric evaluation estimator. Review of Economic Studies, 65(2), 261–294.

  8. Heckman, J. J., Ichimura, H., & Todd, P. E. (1997). Matching as an econometric evaluation estimator: Evidence from evaluating a job training programme. The Review of Economic Studies, 64(4), 605–654.

  9. Hirano, K., Imbens, G. W., & Ridder, G. (2003). Efficient estimation of average treatment effects using the estimated propensity score. Econometrica, 71(4), 1161–1189.

  10. Ho, D. E., Imai, K., King, G., & Stuart, E. A. (2007). Matching as nonparametric preprocessing for reducing model dependence in parametric causal inference. Political Analysis, 15(03), 199–236.

  11. Hoeffding, W. (1963). Probability inequalities for sums of bounded random variables. Journal of the American Statistical Association, 58(301), 13–30.
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  12. Rosenbaum, P. R. (1989). Optimal matching for observational studies. Journal of the American Statistical Association, 84(408), 1024–1032.
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  13. Rosenbaum, P. R. & Rubin, D. B. (1983). The central role of the propensity score in observational studies for causal effects. Biometrika, 70(1), 41–55. A Proofs A.1 Miscellaneous definitions and propositions Definition A1. Let Πsupp be the support of Π. Definition A2. Let Π− supp = inf Πsupp and Π+ supp = sup Πsupp. Definition A3. Let z(p) = E[Y (z)|Π = p] be the conditional expectation of the potential outcome for treatment z given propensity score p.
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  14. Theorem A1. Let X1, X2, . . . , Xn be n independent random variables such that 0 ≤ Xi ≤ 1 with probability one. Let X̄ = n−1 Pn i=1 Xi and = E[X̄]. For 0 < t < 1− and 0 < s < : Pr(X̄ − ≥ t) ≤ exp(−2nt2 ) and Pr(X̄ − ≤ −s) ≤ exp(−2ns2 ). Proof. The first inequality is Theorem 1 in Hoeffding (1963). For the second inequality: Pr(X̄ − ≤ −s) = Pr(−X̄ + ≥ s) = Pr (1 − X̄) − (1 − ) ≥ s , which is bounded by exp(−2ns2 ) when 0 < s < 1 − (1 − ) using the first inequality. Theorem A2. Under Condition 1, Y (0) is conditionally independent of Z given Π = p on the support of Π.
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